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%I #32 Feb 06 2024 08:07:15
%S 1,0,10,0,270,240,10900,25200,551950,2116800,32458860,169092000,
%T 2120787900,13427013600,149506414200,1075081207200,11143223412750,
%U 87198375264000,865743970019500,7171730187336000,69416724049550020
%N Expansion of e.g.f. I_0(2*x)^5 + 2*Sum_{k>=1} I_k(2*x)^5, where I_n(z) are modified Bessel functions of order n.
%C A modification of e.g.f. of A002898, where the exponent of I, which is 3, is here replaced by 5.
%C U_10(n), in Labelle-Lacasse paper, number of closed paths of length n whose steps are 10th roots of unity.
%H Andrew Howroyd, <a href="/A070190/b070190.txt">Table of n, a(n) for n = 0..200</a>
%H Gilbert Labelle and Annie Lacasse, <a href="https://doi.org/10.46298/dmtcs.2937">Closed paths whose steps are roots of unity</a>, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.
%F a(n) ~ 5^(3/2) * 10^n / (4*Pi^2*n^2). - _Vaclav Kotesovec_, Jun 08 2021
%t With[{nmax = 25}, CoefficientList[Series[BesselI[0, 2*x]^5 + 2*Sum[BesselI[k, 2*x]^5, {k, 1, 2*nmax}], {x, 0, nmax}], x]*Range[0, nmax]!] (* _G. C. Greubel_, Nov 05 2018 *)
%o (PARI) seq(n)={Vec(serlaplace(sum(k=0, n, if(k,2,1)*(x^k*besseli(k, 2*x + O(x^(n-k+1)))/k!)^5)))} \\ _Andrew Howroyd_, Nov 01 2018
%Y Cf. A002898.
%K nonn
%O 0,3
%A _Karol A. Penson_, Apr 26 2002
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